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Integer and inequality

Source: China TST 2006 (1)

March 24, 2006
inequalitiesfunctioninequalities unsolved

Problem Statement

Given nn real numbers a1a_1, a2a_2 …\ldots ana_n. (n≥1n\geq 1). Prove that there exists real numbers b1b_1, b2b_2 …\ldots bnb_n satisfying: (a) For any 1≤i≤n1 \leq i \leq n, ai−bia_i - b_i is a positive integer. (b)∑1≤i<j≤n(bi−bj)2≤n2−112\sum_{1 \leq i < j \leq n} (b_i - b_j)^2 \leq \frac{n^2-1}{12}
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